| L(s) = 1 | + 2.41·3-s − 2.02·5-s − 4.66·7-s + 2.83·9-s − 2.63·11-s + 3.53·13-s − 4.87·15-s − 5.86·17-s − 8.36·19-s − 11.2·21-s + 2.50·23-s − 0.918·25-s − 0.400·27-s − 0.112·29-s + 7.97·31-s − 6.35·33-s + 9.42·35-s + 2.28·37-s + 8.54·39-s − 9.21·41-s − 0.720·43-s − 5.72·45-s + 12.0·47-s + 14.7·49-s − 14.1·51-s + 3.14·53-s + 5.31·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.903·5-s − 1.76·7-s + 0.944·9-s − 0.793·11-s + 0.980·13-s − 1.25·15-s − 1.42·17-s − 1.91·19-s − 2.45·21-s + 0.523·23-s − 0.183·25-s − 0.0770·27-s − 0.0208·29-s + 1.43·31-s − 1.10·33-s + 1.59·35-s + 0.375·37-s + 1.36·39-s − 1.43·41-s − 0.109·43-s − 0.853·45-s + 1.76·47-s + 2.10·49-s − 1.98·51-s + 0.432·53-s + 0.716·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315908028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315908028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 8.36T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 + 0.112T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 0.720T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 9.84T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 2.14T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.048448188070696285147780974718, −7.16687063025880958056911985319, −6.56402125206116070425481812860, −6.05615348073109614177200906398, −4.73830754241280914762754951278, −3.81092368111795622421358195297, −3.67886823585995114691958842134, −2.63771436548030154651952495665, −2.26452421917164273641331858167, −0.48934455713687195077598952946,
0.48934455713687195077598952946, 2.26452421917164273641331858167, 2.63771436548030154651952495665, 3.67886823585995114691958842134, 3.81092368111795622421358195297, 4.73830754241280914762754951278, 6.05615348073109614177200906398, 6.56402125206116070425481812860, 7.16687063025880958056911985319, 8.048448188070696285147780974718
